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Chapter 4: Q13SE (page 273)

Suppose thatXandYare random variables for whichE(X)=3,E(Y)=1, Var(X)=4, and Var(Y )=9. LetZ=5X鈭�Y+15. FindE(Z)and Var(Z)under each of thefollowing conditions:

(a)XandYare independent;

(b)XandYare uncorrelated;

(c) the correlation ofXandYis 0.25.

Short Answer

Expert verified
  1. When X and Yare independent, E(Z)=29 AND Var(Z)=109
  2. When X and Yare uncorrelated, E(Z)=29 AND Var(Z)=109
  3. When the correlation betweenX and Yare 0.25, E(Z)=29 AND Var(Z)=94

Step by step solution

01

Given information

X and Y are two random variables. The mean and variance of that two variables are,

\(\begin{aligned}{}E\left( X \right) = 3\\E\left( Y \right) = 1\\Var\left( X \right) = 4\\Var\left( Y \right) = 9\end{aligned}\)

Define another random variable \(Z = 5X - Y + 15\).

02

Calculate mean and variance when independent

a.

When Xand Y are independent, the covariance between them is 0 and correlation between them is also 0.

So, the mean of Z is,

\(\begin{aligned}{}E\left( Z \right) = E\left( {5X - Y + 15} \right)\\ = 5E\left( X \right) - E\left( Y \right) + 15\\ = 15 - 1 + 15\\ = 29\end{aligned}\)

The variance of Z is,

\(\begin{aligned}{}Var\left( Z \right) = {5^2}Var\left( X \right) + {\left( { - 1} \right)^2}Var\left( Y \right) + 2 \times 5 \times \left( { - 1} \right)\sqrt {Var\left( X \right)Var\left( Y \right)Cor\left( {X,Y} \right)} \\ = 25Var\left( X \right) + Var\left( Y \right) - 0\\ = 25 \times 4 + 9\\ = 109\end{aligned}\)

03

Calculate mean and variance when uncorrelated

b.

When X and Y are uncorrelated then the covariance between them is 0.

So, the mean of Z is,

\(\begin{aligned}{}E\left( Z \right) = E\left( {5X - Y + 15} \right)\\ = 5E\left( X \right) - E\left( Y \right) + 15\\ = 15 - 1 + 15\\ = 29\end{aligned}\)

The variance of Z is,

\(\begin{aligned}{}Var\left( Z \right) = {5^2}Var\left( X \right) + {\left( { - 1} \right)^2}Var\left( Y \right) + 2 \times 5 \times \left( { - 1} \right)\sqrt {Var\left( X \right)Var\left( Y \right)Cor\left( {X,Y} \right)} \\ = 25Var\left( X \right) + Var\left( Y \right) - 0\\ = 25 \times 4 + 9\\ = 109\end{aligned}\)

04

Calculate mean and variance when correlation is 0.25

c.

Here the correlation between X and Y is 0.25

So, the mean of Z is,

\(\begin{aligned}{}E\left( Z \right) = E\left( {5X - Y + 15} \right)\\ = 5E\left( X \right) - E\left( Y \right) + 15\\ = 15 - 1 + 15\\ = 29\end{aligned}\)

The variance of Z is,

\(\begin{aligned}{}Var\left( Z \right) = {5^2}Var\left( X \right) + {\left( { - 1} \right)^2}Var\left( Y \right) + 2 \times 5 \times \left( { - 1} \right)\sqrt {Var\left( X \right)Var\left( Y \right)Cor\left( {X,Y} \right)} \\ = 25Var\left( X \right) + Var\left( Y \right) - 10 \times 2 \times 3 \times 0.25\\ = 25 \times 4 + 9 - 15\\ = 94\end{aligned}\)

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